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n divisible by 6

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j_shreyans GMAT Destroyer! Default Avatar
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n divisible by 6 Post Wed Sep 10, 2014 11:06 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    Is the positive integer n divisible by 6?

    (1) n^2/180 is an integer.

    (2) 144/n^2 is an integer

    OAA

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    GMATinsight GMAT Destroyer!
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    Post Thu Sep 11, 2014 4:12 am
    j_shreyans wrote:
    Is the positive integer n divisible by 6?

    (1) n^2/180 is an integer.

    (2) 144/n^2 is an integer

    OAA
    Question : Is the positive integer n divisible by 6?
    Question : Is n Multiple of 2 as well as 3?

    Statement 1) n^2/180 is an integer.
    i.e. n^2 = 2^2 x 3^2 x 5 x A where A is any Integer
    i.e. n is a multiple of 2 as well as 3
    SUFFICIENT

    Statement 2) 144/n^2 is an integer
    i.e. 2^4 x 3^2 is divisible by n^2

    now it is possible for n = 6 (a multiple of 6
    also it is possible for n = 4 which is not a multiple of 6
    Inconsistent answer therefore
    NOT SUFFICIENT

    Answer: Option A

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    Last edited by GMATinsight on Mon Oct 12, 2015 7:44 pm; edited 1 time in total

    Post Thu Sep 11, 2014 6:35 am
    j_shreyans wrote:
    Is the positive integer n divisible by 6?

    (1) n²/180 is an integer.

    (2) 144/n² is an integer

    OAA
    NOTE: This question is VERY similar to this one: http://www.beatthegmat.com/ds-problem-t279217.html

    Okay, a lot of integer property questions can be solved using prime factorization.
    For questions involving divisibility, divisors, factors and multiples, we can say:
    If N is divisible by k, then k is "hiding" within the prime factorization of N

    Consider these examples:
    24 is divisible by 3 because 24 = (2)(2)(2)(3)
    Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
    And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
    And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)

    For more on this concept, see the 2nd half of this free video: http://www.gmatprepnow.com/module/gmat-integer-properties/video/825

    ------------------------------

    Okay, onto the question!

    Target question: Is the positive integer n divisible by 6?

    Statement 1: n²/180 is an integer
    This tells us that n² is DIVISIBLE by 180
    This means that 180 is "hiding in the prime factorization of n²
    180 = (2)(2)(3)(3)(5)
    So, n² = (2)(2)(3)(3)(5)(?)(?)(?)(?)

    Aside: the (?)'s represent other possible primes in the prime factorization of n²

    Rewrite as (n)(n) = [(2)(3)(5)(?)(?)][(2)(3)(5)(?)(?)]
    This tells us that we can be certain that n = (2)(3)(5)(?)(?)
    At this point it is clear that n is divisible by 6
    Since we can answer the target question with certainty, statement 1 is SUFFICIENT

    Statement 2: 144/n² is an integer
    There are several values of n that satisfy this condition. Here are two:
    Case a: n = 1. Notice that 144/1² = 144, and 144 is an integer. In this case n is NOT divisible by 6
    Case b: n = 6. Notice that 144/6² = 4, and 4 is an integer. In this case n IS divisible by 6
    Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

    Answer = A

    --------------------

    Here are a few more questions that require us to understand the relationship between divisibility and prime factorization:
    http://www.beatthegmat.com/factors-t272765.html
    http://www.beatthegmat.com/do-t-and-12-have-common-factors-t274544.html
    http://www.beatthegmat.com/is-xy-a-multiple-of-105-t274522.html
    http://www.beatthegmat.com/confused-need-help-here-t271655.html
    http://www.beatthegmat.com/multiple-of-990-t272719.html
    http://www.beatthegmat.com/if-n-t-3-for-some-positive-integer-t148420.html

    Cheers,
    Brent

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    prada Rising GMAT Star Default Avatar
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    Post Mon Oct 12, 2015 5:12 pm
    [quote="GMATinsight"]
    j_shreyans wrote:
    Statement 2) 144/n^2 is an integer
    i.e. 2^4 x 3^3 is divisible by n^2


    Answer: Option A
    Sorry for bumping an old thread (linked from GMATPREP) but shouldn't it be 2^4 x 3^2?
    I think you did the PF of 144?

    GMATinsight GMAT Destroyer!
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    Post Mon Oct 12, 2015 7:46 pm
    [quote="prada"]
    GMATinsight wrote:
    j_shreyans wrote:
    Statement 2) 144/n^2 is an integer
    i.e. 2^4 x 3^3 is divisible by n^2


    Answer: Option A
    Sorry for bumping an old thread (linked from GMATPREP) but shouldn't it be 2^4 x 3^2?
    I think you did the PF of 144?
    Yes, It was PF of 144. I made a typo mistake. Thanks for taking note of it. Have made the necessary changes.

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